Numerical Mathematics 1
Faculteit  Science and Engineering 
Jaar  2017/18 
Vakcode  WINM107 
Vaknaam  Numerical Mathematics 1 
Niveau(s)  bachelor 
Voertaal  Engels 
Periode  semester II b 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Numerical Mathematics 1  
Leerdoelen  At the end of the course, the student is able to: 1. describe the methods mentioned in the description. 2. reproduce basic concepts and theorems from Numerical Analysis related to the subjects mentioned in the description. 3. apply techniques from vector analysis and linear algebra to analyze the treated numerical methods. 4. create an algorithm from a given numerical method and to implement it in MATLAB. 5. assess some of the difficulties that may come up when solving a problem on a computer by a specific algorithm, specifically, the difficulties arising from finite precision computations (propagation of roundoff errors), limited memory availability, and finite computation speed (complexity of the algorithm). 6. make a judicious choice from the discussed methods for a particular problem. 

Omschrijving  Most mathematical problems cannot be solved analytically. Numerical mathematics supplies the tools to solve such problems approximately by the computer. In this course we will treat both the theory and algorithms of numerical methods for solving the following problems • Numerical Linear Algebra problems: Solving linear systems: o Direct methods: Condition number, LU factorization, Cholesky factorization, banded systems. o Iterative methods: Classical methods (Jacobi, Gauss Seidel, SOR), (conjugate) Gradient method, preconditioning, stopping criteria. o Eigenvalue problems: the simplest numerical method to compute eigenvalues: the power method. • Nonlinear equations and systems: Aitken extrapolation and error estimation, Stephensen's method, the fixedpoint method, of which (Quasi) Newton methods are special cases. • Interpolation problems The Lagrange interpolating polynomial, the interpolation error (in particular the Runge phenomenon) and the more robust approach by piecewise interpolation. • Integral evaluation: methods based on polynomial interpolation. In particular the (composite) midpoint, trapezoidal and Simpson will be introduced and analyzed. • Ordinary differential equations (initial value problems). One can restrict to methods for firstorder systems. Two important classes will be studied: (i) multistep methods (ii) RungeKutta methods. For these classes zero stability and absolute stability will be studied. • Partial differential equations. Using finite differences (i.e. approximations to derivatives) on a grid. Using this approach algorithms for elliptic (e.g. Poisson equation) and parabolic (e.g. heat equation) will be derived. 

Uren per week  
Onderwijsvorm  Hoorcollege (LC), Practisch werk (PRC), Werkcollege (T)  
Toetsvorm 
Practisch werk (PR), Schriftelijk tentamen (WE)
(There are 6 practicals and 1 written exam. The practicals count for 40% and the exam for 60%. Both the practicals and the exam have to be grade 5.5 or higher. The Lab Sessions are mandatory. Deadlines will be set; not meeting a deadline means 0 points for that practical.) 

Vaksoort  bachelor en master  
Coördinator  dr. C.A. Bertoglio  
Docent(en)  dr. C.A. Bertoglio  
Verplichte literatuur 


Entreevoorwaarden  Background knowledge and skill in handling basic numerical techniques and programming in Matlab. In particular the student should be able to work with functions and to pass an anonymous function as a parameter. Basic knowledge of linear algebra (eigenvalues, eigenvectors, norms) and calculus (Tayloer series of multivariate functions) is required.  
Opmerkingen  This course is a mandatory prerequisite for the courses Computational Methods of Science, Numerical Mathematics 2, in general for all more advanced Numerical Mathematics courses.  
Opgenomen in 
